According to physicalism, everything is physical or metaphysically connected to the physical. If physicalism were true, it seems that we should - in principle - be able to reduce the descriptions and explanations of special sciences to physical ones, for example, explaining biological regularities, via chemistry, by the laws of particle physics. The multiple realization of the property types of the special sciences is often seen to be an obstacle to such epistemic reductions. Here, we introduce another, new argument against epistemic reduction. Based on mathematical complexity, we show that, under certain conditions, there can be "complexity barriers" that make epistemic reduction - in principle - unachievable even if physicalism were true.